diff --git a/src/new_autgroups.jl b/src/new_autgroups.jl
index 4c628a0..4be666a 100644
--- a/src/new_autgroups.jl
+++ b/src/new_autgroups.jl
@@ -33,7 +33,7 @@ function relations(G::AutomorphismGroup)
     return last(gersten_relations(n, commutative=false))
 end
 
-_hashing_data(f::FPGroupElement{<:AutomorphismGroup}) = normalform!.(evaluate(f))
+equality_data(f::FPGroupElement{<:AutomorphismGroup}) = normalform!.(evaluate(f))
 
 function Base.:(==)(g::A, h::A) where A<:FPGroupElement{<:AutomorphismGroup}
     @assert parent(g) === parent(h)
@@ -52,12 +52,12 @@ function Base.:(==)(g::A, h::A) where A<:FPGroupElement{<:AutomorphismGroup}
     img_computed, imh_computed = false, false
 
     if !_isvalidhash(g)
-        img = _hashing_data(g)
+        img = equality_data(g)
         _update_savedhash!(g, img)
         img_computed = true
     end
     if !_isvalidhash(h)
-        imh = _hashing_data(h)
+        imh = equality_data(h)
         _update_savedhash!(h, imh)
         imh_computed = true
     end
@@ -68,17 +68,15 @@ function Base.:(==)(g::A, h::A) where A<:FPGroupElement{<:AutomorphismGroup}
     hash(g) != hash(h) && return false
 
     # words are different, but hashes agree
-    if !img_computed
-        img = _hashing_data(g)
-    end
-    if !imh_computed
-        imh = _hashing_data(h)
+    @sync begin
+        !img_computed && Threads.@spawn img = equality_data(g)
+        !imh_computed && Threads.@spawn imh = equality_data(h)
     end
 
-    res = img == imh
-    !res && @warn "hash collision in == :" g h
+    equal = img == imh
+    equal || @warn "hash collision in == :" g h
 
-    return res
+    return equal
 end
 
 # eye-candy
diff --git a/src/new_hashing.jl b/src/new_hashing.jl
index 20dceab..3c771c2 100644
--- a/src/new_hashing.jl
+++ b/src/new_hashing.jl
@@ -1,6 +1,6 @@
 ## Hashing
 
-_hashing_data(g::FPGroupElement) = word(g)
+equality_data(g::FPGroupElement) = (normalform!(g); word(g))
 
 bitget(h::UInt, n::Int) = Bool((h & (1 << n)) >> n)
 bitclear(h::UInt, n::Int) = h & ~(1 << n)
@@ -27,7 +27,7 @@ _setvalidhash!(g::FPGroupElement, v::Bool) =
 
 # To update hash use this internal method, possibly only after computing the
 # normal form of `g`:
-function _update_savedhash!(g::FPGroupElement, data=_hashing_data(g))
+function _update_savedhash!(g::FPGroupElement, data)
     h = hash(data, hash(parent(g)))
     h = (h << count_ones(__BITFLAGS_MASK)) | (__BITFLAGS_MASK & g.savedhash)
     g.savedhash = _setvalidhash(h, true)
@@ -36,7 +36,7 @@ end
 
 function Base.hash(g::FPGroupElement, h::UInt)
     normalform!(g)
-    _isvalidhash(g) || _update_savedhash!(g)
+    _isvalidhash(g) || _update_savedhash!(g, equality_data(g))
     return hash(g.savedhash >> count_ones(__BITFLAGS_MASK) , h)
 end